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Unformatted text preview: MAT 485 (002) — FINAL EXAM Name : May 6, 2003
Time : 2 hours. Total Marks: 100
Instructions 1. Attempt all questions. 2. Show all your steps, answers without justiﬁcation will not receive full credit. A mere ’yes’ or ’no’ is not acceptable. 3. This exam contains IX problems and 13 pages including a cover sheet. Make sure you have all the sheets. DO NOT WRITE BELOW THIS LINE I. II.
III. IV.
V. VI.
VII. VIII.
IX. TOTAL MAT 485 (002)  FINAL EXAM I. Solve the following system of equations using Gauss or Gauss Jordan elimination. Show all the elementary row operations you are using. (10 marks) x+y=3
y+z=4 x+z=5 MAT 485 (002)  FINAL EXAM 3 Ila. Are the vectors (2,—4, —6), (—3,5,7),(—4, 5,6) linearly inde—
pendent in R3? Justify your answer. (6 marks) 11 b. Are the functions f1(t) = 1,f2(t) : e‘t, f3(t) = t6_t linearly independent? Justify your answer. (6 marks) MAT 485 (002)  FINAL EXAM III. Solve the following initial value problem by separating the vari
ables
y3y' + $3 = 0, y(0) =1
(6 marks) MAT 485 (002) — FINAL EXAM 5 IV. Consider the differential equation
(cosy + ycosm)dx + (sinx — xsiny)dy 2 0. (a) Is the equation exact? Justify your answer. (4 marks) (b) Find the general solution to the above differential equa— tion. (6 marks) MAT 485 (002) — FINAL EXAM V. A tank contains 200 gallons of fresh water. A solution contain—
ing 2 lb/gal of salt is pumped into the tank at 5 gal/min. The
mixture is stirred constantly and ﬂows out at the same rate of
5gal/ min. (a) What is the actual amount of salt y(t) at time t? (b) How much salt is in the tank after 10 minutes? (12 marks) MAT 485 (002)  FINAL EXAM Extra page for problem V. MAT 485 (002)  FINAL EXAM VI. Solve the IVP for the following Euler—Cauchy equation:
9321/” — 123/ + 5y = 0, y(1) = 2, y’(1)= 0 (10 marks) MAT 485 (002)  FINAL EXAM 9 VII. Solve the following IVP.
y” + 3y’ + 2y = 36”, y(0) = 0, y’(0) = 0 (15 marks) 10 MAT 485 (002)  FINAL EXAM Extra page for problem VII. MAT 485 (002)  FINAL EXAM 11
VIII. Find a solution to the following system of linear differential
equations:
yi = 61/1 — 312
y; = 5y1 + 4y2 (15 marks) 12 MAT 485 (002) — FINAL EXAM Extra page for problem VIII. MAT 485 (002)  FINAL EXAM 13 IX. Find a power series solution in powers of :1: to the following differential equation (Show the details of your work):
(1  332W 2 2w (10 marks) ...
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This note was uploaded on 01/15/2012 for the course MAT 485 taught by Professor Staff during the Fall '11 term at Syracuse.
 Fall '11
 Staff

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